Electrostatic fields

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Electrostatic fields

• Sandra Cruz-Pol, Ph. D.
• INEL 4151 ch 4
• ECE UPRM
• Mayagüez, PR

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Some applications

• Power transmission, X rays, lightning protection
• Solid-state Electronics resistors, capacitors,
  FET
• Computer peripherals touch pads, LCD, CRT
• Medicine electrocardiograms, electroencephalogram
  s, monitoring eye activity
• Agriculture seed sorting, moisture content
  monitoring, spinning cotton,
• Art spray painting
• many more!

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We will study Electric charges

• Coulomb's Law
• Use when charge distribution is known
• Gausss Law
• Use when charge distribution is symmetrical
• Electric Potential, V
• (uses scalar, not vectors)
• Use when potential V is known

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Coulombs Law (1785)

• Force one charge exerts on another
• where k 9 x 109
• or k 1/4peo
• e08.85 x 10-12

R


Superposition applies
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Force with direction
Force that Q1 exerts on Q2
Note Observation point goes First!
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Example

• If two point charges 5nC and -2nC are located at
  r1(2,0,4) and r2(-3,0,5), respectively.
• Find the force on a 1nC point charge, Qx,
  located at (1,-3,7)

Apply superposition
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Electric field intensity, E

• Is the force per unit charge when placed in the E
  field

Example Same point charges 5nC and -2nC are
located at (2,0,4) and (-3,0,5), respectively.
b) Find the E field at rx(1,-3,7).
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If we have many charges
Line charge density, rL C/m
Surface charge density rS C/m2
Volume charge density rv C/m3
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The total E-field intensity is
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More Charge distributions

• Find E from
• Point charge (we just saw this one)
• Line charge
• Surface charge
• Volume charge

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Results Preview
Spoiler Alert!?

• We will derive these 3 cases
• Using Coulomb
• Using Gauss

Line charge Sheet charge Volume Charge (same as
Point Charge but Q is total within volume)
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Find E from LINE charge

• Line charge w/uniform charge density, rL
• use cylindrical coordinates

z
(x,y,z)
dE
T
Observation Point
B
R
(0,0, z)
dl
A
x
O
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FIRST Define angles a1 and a2

• a1 imaginary perpendicular line with the back
  (tail)
• a2imaginary perpendicular line with the front
  (head)

z
dE
T
a 2
B
a 1
dl
A
x
0
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LINE charge

• Substituting in

z
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examples
a 2?
a 2
a 1
a 1
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examples
a 2
a 1
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Results Preview
Line charge Sheet charge Volume Charge
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More Charge distributions

• Point charge (same eq as for Volume Charge)
• Line charge
• Surface charge
• Volume charge

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Find E from Surface charge

• Sheet of charge with uniform density rS

Element of area is
z
Observation point is at z-axis
y
Substituting all
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SURFACE charge

• Due to SYMMETRY
• the r component cancels out.

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Results Preview
Line charge Sheet charge Volume Charge
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More Charge distributions

• Point charge
• Line charge
• Surface charge
• Volume charge

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Find E from Volume charge

• Sphere of charge w/uniform density, rv

P(0,0,z)
dE
R
a
(Eq. )
z
(r,q,f)
q
rv
Differentiating (Eq. )
f
x
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Find E from Volume charge

• Substituting

dE
P(0,0,z)
R
(r,q,f)
rv
q
f
x
Where do limits on R come from?
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Results Preview
Line charge Sheet charge Volume Charge
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P.E. 4.5

• A square plate at plane z0
  and carries a charge
  mC/m2 . Find
• the total charge on the plate and
• the electric field intensity at (0,0,10).

10 --

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Cont
z 10
sheet of charge
y 2
x 2
Due to symmetry only Ez survives
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Chapter Outline

• Coulomb's Law-
• Use when charge distribution is known
• Gausss Law
• Use when charge distribution is symmetrical
• Electric Potential
• (uses scalar, not vectors)
• Use when potential V is known

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Gausss Law
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Review

• How to draw the xyz coordinate system

z
y
x
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Define Y DElectric Flux Density
First
D is independent of the medium in which the
charge is placed.
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Gausss Law- derivation
Note the area S is closed
Therefore
This is the 1st of the Maxwells equations
derived here.
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Gausss Law-description

• The total electric flux Y, through any closed
  surface is equal to the total charge enclosed by
  that surface.

The key is to choose the Gauss surface, S, to
make the Dot Product1. Look at the symmetry of
the particular case. Pick surface so that D is
to it.
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Results (same w/Gauss)
Point charge Line charge Sheet charge Volume
Charge
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Point Charge Finding D at point P from the
charges

• Point Charge is at the origin.
• Choose a spherical dS
• Note where D is perpendicular to this surface.

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Results (same w/Gauss)
Line charge Sheet charge Volume Charge (point
charge Q)
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Line Charge Finding D at point P from the
charges

• Infinite Line Charge
• Choose a cylindrical dS
• Note that integral 0 at top and bottom surfaces
  of cylinder

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Results (same w/Gauss)
Line charge Sheet charge Volume Charge
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Sheet of charge Find D at point P from the
charges

• Infinite Sheet of charge
• Choose a cylindrical box cutting the sheet
• Note that D is parallel to the sides of the box.

sheet of charge
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Results (same w/Gauss)
Line charge Sheet charge Volume Charge
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P.E. 4.7

• A point charge of 30nC is located at the
  origin, while plane y 3 carries charge 10nC/m2.
• Find D at (0, 4, 3)

z
y
30nC
10nC/m2
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Problem 4.17.

• A 12nC charge is placed at the origin. Determine
  the electric flux passing through (a) the surface
  defined by
• r 1, 0ltqlt60o,
• 0lt f lt 2p.
• Draw the problem
• Ans 3n C

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P.E. 4.8

• If
  C/m2 .
• Find
• volume charge density at (-1,0,3)
• Flux thru a cube defined by
• Total charge enclosed by the cube

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4.23 Find D everywhere

• given that

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Ex. Find D everywhere

• Given that

5
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Chapter Outline

• Coulomb's Law-
• Use when charge distribution is known
• Gausss Law
• Use when charge distribution is symmetrical
• Electric Potential , V
• (uses scalar, not vectors)
• Use when potential V is known

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Electric Potential, V

• The work done to move a charge Q from A to B is
• The (-) means the work is done by an external
  force.
• The total work potential energy required in
  moving Q
• The energy per unit charge potential difference
  between the 2 points

V is independent of the path taken.
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• The Potential at any point is the potential
  difference between that point and a chosen
  reference point at which the potential is zero.
  (choosing infinity)
• For many Point charges at rk
• (apply superposition)
• For Line Charges
• For Surface charges
• For Volume charges

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P.E. 4.10

• A point charge of -4mC is located at (2,-1,3)
• A point charge of 5mC is located at (0,4,-2)
• A point charge of 3mC is located at the origin
• Assume V(8)0 and Find the potential at (-1, 5,
  2)

gt C0
10.23 kV
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P.E. 4.11

• A point charge of 5nC is located at the origin
  V(0,6,-8)2V and Find the potential at point
  A(-3, 2, 6)
• Find the potential at B(1,5,7), the potential
  difference VAB

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Example

• A line charge of 5nC/m is located on a line at
  x10, y 20
• Assume V(0,0,0)0 and Find the potential at A(3,
  0, 5)

VA4.8V
r0(0,0,0)-(10, 20, 0)22.36 and
rA(3,0,5)-(10, 20, 5) 21.2
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Relation between E and V
V is independent of the path taken.
B
Esto aplica sólo a campos estáticos. Significa
que no hay trabajo NETO en mover una carga en un
paso cerrado donde haya un campo estático E.
Apply Stokes Theorem
A
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Static E satisfies
B
Condition for Conservative field independent of
path of integration
A
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Example

• Given the potential
• Find D at .

In spherical coordinates (r,q,f)